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\Title About  Mechanically Generated Curves.
\medskip

\lf
\cl{Examples}\lf
We have the following mechanically generated plane curves
programmed,  together with a decoration that shows this generation
and the corresponding construction of the tangents of the curve:

%\includegraphics[width=1.2in] {Mechanically_Generated_Curves.png}
\noindent
\hbox{ 
\vbox{\hsize=0.4\hsize \phantom{.}\hskip-8mm
\includegraphics[width=1.2in] {Mechanically_Generated_Curves.png}}
\vbox{\hsize=0.5\hsize \noindent
Epi- and Hypocycloids, \lf all other rolling curves.
\lf
Also: Tractrix, Cissoid, \lf Conchoid, Lemniscate. \lf
}}

\noindent
This image is obtained with {\tt Color Morph} in the Animation menu, it shows
the family obtained from the current drawing mechanism (here Lemniscate).
\Lf
\cl{Moving Planes}\lf
It is often convenient to discuss such mechanical generations in terms
of two planes, a fixed plane on which the drawing is done (paper plane)
and a second plane which is attached to that piece of the mechanical
contraption that holds the drawing pen (drawing plane). In the case of
rolling curves we have the drawing plane attached to the rolling wheel,
in the case of the Lemniscate the drawing plane is attached to the middle
one of the three connected moving segments.
\par{\narrower \noindent  \it 
 We think of the orbits of the points of the drawing plane as curves that are 
mechanically generated by the apparatus under consideration.\par}
\noindent
The velocity vectors of these orbits clearly give a time dependent vector field.
Since this vector field is obtained by differentiating the orbits of a family of
{\bf iso\-metries} we obtain at each time t the vector field of a Euclidean
{\bf group of motions}, in other words: for most t the vector field consists of
the velocity vectors of a rotation, a rotation around the so called momentary center
of rotation. This way of looking at the generation gives immediately tangent
constructions for all orbits: join the momentary center of rotation to the moving
point, the perpendicular line through the point is tangent to its orbit.
\lf
It is therefore useful to visualize the movement of the drawing plane together
with the time dependent velocity field of its points. We have done this by 
decorating the drawing plane with not too many but enough random points
so that the movement of the drawing plane becomes visible, but the curve
under consideration is not obscured. Moreover, to make the vector field
visible at each moment t, we have drawn the random points not once, but
at two subsequent positions. This picture is interpreted by the brain correctly.
\lf
Finally, one has to determine the momentary center of rotation. This is different
for each construction. For rolling curves the definition of ``rolling'' is such that
that point, where the rolling wheel touches the fixed curve (``street''), is the
momentary center of rotation. In general one has to look for points of the
mechanical apparatus for which the direction of the momentary movement
(``orbit tangent'') can be decided. The momentary center is then on the line 
(``radius'') perpendicular to the tangent, so that two such lines are needed. 
The 3DXM demos use green lines to determine the momentary center.
\LF
H.K.


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